Questions tagged [complex-manifolds]
For questions about complex manifolds.
333
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Compact connected complex/holomorphic manifold that embeds in $\mathbb{C}^n$
If $i : M \to \mathbb{C}^n$ is a holomorphic map, then each coordinate function on $\mathbb{C}^n$ restricts to a global holomorphic function on the image. In particular, there is no holomorphic ...
- 35
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0
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42
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Analytic variety induced by an irreducible polynomial
I am reading Principles of Algebraic Geometry by Griffith and Harris. Here the authors define an analytic variety in a domain as follows :
A subset $V$ of an open set $U \subset \mathbb{C}^n$ is an ...
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0
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43
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Integrable almost complex structure on a torus
Consider the torus $T^2 = S^1 \times S^1$ and let $(x,y)$ be the canonical coordinates $(0<x<2\pi, 0<y<2\pi)$ on $T^2$. The corresponding coordinate vector fields define global fields ...
0
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31
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Holomorphic maps of complex manifolds preserve the bidegree of a complex differential form
For a holomorphic map $f : M \to N$ between complex manifolds show that if $\omega$ is a form of type $(p,q)$ on $N$, then $f^*\omega$ is a form of type $(p,q)$ on $M$.
I am wondering if the ...
- 309
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35
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Show that $X$ admits a natural almost complex structure and that any almost complex structure on $X$ is induced by a complex structure.
Let $X$ be an oriented Riemann surface. Show that $X$ admits a natural almost complex structure and that any almost complex structure on $X$ is induced by a complex structure.
Okay we want to define $...
- 309
3
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1
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76
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Redundancy with basis vectors for $T_pM^\mathbb C \cong T_pM^{1,0} \oplus T_pM^{0,1}$
I've found out that the primary motivation for complexifying the tangent bundle is to enable its decomposition into the eigenspaces $\pm i$ of $J$. This decomposition facilitates the splitting of the ...
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73
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Basis for $T_pM^\mathbb{C}$
For a real vector space $V$ with dimension $n$ and a basis $\{e_1,\dots,e_n\}$ the complexification $V^\mathbb{C}=V \otimes \mathbb C$ contains of vectors of the form $v+iw$ for $v,w\in V$. Now write $...
- 203
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40
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Show that any non-trivial homogeneous polynomial $s$ of degree $k$ can be considered as a non-trivial section of $\mathcal{O}(k)$ on $\mathbb{P}^n$.
Show that any non-trivial homogeneous polynomial $0 \ne s \in \mathbb{C}[z_0, \dots, z_n]$ of degree $k$ can be considered as a non-trivial section of $\mathcal{O}(k)$ on $\mathbb{P}^n$.
So $\mathcal{...
- 309
1
vote
1
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96
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Structure on a complex manifold and the Kähler form on $\mathbb{C}^2$.
Currently learning about complex and Kähler manifolds and I'm reading this article about Kähler forms.
Specifically I'm trying to understand how do they conclude that on $\mathbb{C}^2$ the Kähler form ...
- 561
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52
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"Degree function" of holomorphic map between compact Riemann surfaces
I am currently reading the proof about the degree of a holomorphic (nonconstant) function between compact Riemann surfaces in "Algebraic Curves and Riemann Surfaces" by R. Miranda.
The ...
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0
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18
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The Fibrewise Multiplication on a Normal Vector Bundle of a subset of a Compact Kähler Surface
Let $\mathbb{A}$ be a (non-empty) subset of a compact kähler surface $\mathbb{X}$.
Denote by $\mathcal{N}(\mathbb{A})$ the normal vector bundle of $\mathbb{A}$.
What do they mean by saying "let $\...
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47
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Proof of the fact that $\mathfrak{g}\otimes C^{\infty}(S^1) \cong C^{\infty}(S^1;\mathfrak{g})$
I encountered the following fact: let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$, and consider the tensor product $\mathfrak{g}\otimes_{\mathbb{C}} C^{\infty}(S^1)$ (where I guess we mean maps ...
- 41
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Proof of Spherical metric on Riemann Sphere
While studying stereographic projection of extended complex on unit sphere $S$ in $\mathbb{R^3}$ we get two metrics one is chordal metric and second one is spherical metric. The spherical metric $d_s$ ...
- 57
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40
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Definition of meromorphic function between complex manifolds
Ususally we only consider meromorphic function from a complex manifold $X$ to $\mathbb{C}$:
Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is sheaf of holomorphic functions. We ...
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What is the definition of sheaf of meromorphic differential form on a complex manifold？
Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is the sheaf of $\mathbb{C}$-valued holomorphic functions on $X$.
Let $T_X^\vee$ be the cotangent bundle over $X$ with the ...
- 1,547
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0
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14
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fibres varying non-holomorphically
I'm currently reading Claire Voisin's book Hodge theory and complex manifolds. Here is an extract from it:
in the last paragraph she writes that we cannot chose the trivialisation $T$ to be ...
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48
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$f:M \rightarrow N$ holomorphic between equidimensional complex manifolds is surjective if $|J(f)| \not\equiv 0$
In the book "Principles of algebraic geometry" by Griffiths and Harris (PG. 237) there is a proof of the following statement:
"Let $f:M \rightarrow N$ be a holomorphic map between two ...
- 3
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1
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96
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Partial derivatives chart-dependent?
Let $X$ be a Riemann surface. Let $Y\subset X$ be open.
The following definitions are taken from page 60 of Forster's Riemann Surfaces.
We call a function $f\colon Y \rightarrow \mathbb{C}$ (...
- 601
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1
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128
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Singular irreducible affine plane curve is never a Riemann surface?
Let $f\in \mathbb{C}[x,z]$ be a bivariate irreducible polynomial over the complex numbers.
In case that $f$ is non-singular, one can endow the locus (zero set) of $f$ (considered with subspace ...
- 601
1
vote
1
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97
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Multiplicity of a non-constant holomorphic map
In Miranda's Algebraic Curves and Riemann Surfaces he defines the multiplicity of a non-constant holomorphic map $F\colon X \rightarrow Y$ between Riemann surfaces as the unique integer $m$ such that ...
- 601
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25
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Source for nomenclature "metric tensor" on Hermitian manifold
I am a wikipedia editor and am looking for reliable sources to cite for the term metric tensor in the context of Hilbert spaces or Hermitian manifolds.
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31
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Expressing "the current $T$ gives no mass to the subset $E$" in terms of differential forms on the Complex Projective Plane $\mathbb{CP}^2$
Let $(\mathbb{CP}^2, \omega)$ be the Complex Projective Plane, where $\omega$ is a Hermitian Metric (or, the Kähler Form).
Let $D^{(1,1)}(\mathbb{CP}^2)$ be the space of $(1,1)$-differential forms on ...
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Holomorphic forms are closed on compact manifold $X$ if $\dim(X)=2$.
Let $X$ be a compact complex manifold and $\dim(X)=2$, $\eta$ is a holomorphic form on $X$. Prove that d$\eta=0$.
I know when $X$ is a compact complex Kähler manifold, holomorphic forms are closed.
In ...
- 309
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0
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96
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Chern class of a hypersurface in $\mathbb{C}P^3$
Let $X=\{[z_0,z_1,z_2,z_3]\ \big{|}\ [z_0,z_1,z_2,z_3]\in\mathbb{C}P^3,z_0^4+z_1^4+z_2^4+z_3^4=0\}$. $c_1(X)$ is the first Chern class of $X$. Prove that $c_1(X)=0$.
$\textbf{My try}$:
It's easy to ...
- 309
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1
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66
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affine Gauss map of Hypersurface Manifold finite
Let $M \subset \mathbb{C}^n $ be an algebraic $n-1$-dimensional manifold given as vanishing hypersurface of a polynomial $F \in \mathbb{C}[x_1,..., x_n]$ of degree $d \ge 2$. The smoothness of $M$ can ...
8
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Is a finite covering of a $\partial\bar{\partial}$-manifold still $\partial\bar{\partial}$-manifold?
A compact complex manifold is called a $\partial\bar{\partial}$-manifold if for every pair $p,q\in \mathbb{N}$, every smooth $d$-closed $(p,q)$-form $\eta$ on $X$, $\eta$ is $d$-exact iff $\partial$-...
- 1,020
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43
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Decompose a finite (not necessarily positive) measure $\nu$ into two mutually singular measures $\lambda_1$ and $\lambda_2$
Let $\nu$ be a non-zero finite (not necessarily positive) measure on a compact Kähler surface $M$.
Is it always possible to decompose the measure $\nu$ as follows $\nu=\lambda_1-\lambda_2$?
Where $\...
- 2,430
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21
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Maximal index of a handle in Weinstein manifold/domain.
Consider Weinstein domain $(W,\omega,X,f )$, that is an exact
symplectic manifold with boundary, equipped with vector field X and a
morse function such that:
X is transverse to $\partial W$ pointing ...
- 121
1
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2
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370
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Is the Riemann Surface really a complex-manifold? Or is it without a point 0?
I just learnt basic complex analysis in school. My teacher simply described the Riemann Surface to explain the multi-valued functions better. I know that the so-called "multi-valued functions&...
- 19
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64
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The Self-Intersection Number of the Complex Projective Line $\mathbb{C}\mathbb{P}^1$
Let $\mathbb{C}\mathbb{P}^2$ be the Complex Projective Plane.
Let $C \subset \mathbb{C}\mathbb{P}^2$ be a Rational Curve, i.e., $C$ is isomorphic to the Complex Projective Line $\mathbb{C}\mathbb{P}^1$...
- 2,430
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The Definition of the Rational Curve $C$ Contained in the Complex Projective Plane $\mathbb{C}\mathbb{P}^2$
Let $\mathbb{C}\mathbb{P}^2$ be the Complex Projective Plane.
What do we mean by saying that "$C$ is a Rational Curve contained in $\mathbb{C}\mathbb{P}^2$"?
What does "Rational Curve&...
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1
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1
answer
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the features of the action of $\Gamma = \langle \gamma \rangle$ on $\mathbb{C}\mathbb{P}^1$
Let $\gamma$ be an Elliptic element of ${\rm PSL}(2,\mathbb{C})$ representing an Irrational rotation.
Let $\Gamma$ be the subgroup of ${\rm PSL}(2,\mathbb{C})$ generated by $\gamma$ (i.e., $\Gamma = \...
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1
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Regarding the Defintion of Hirzebruch Surfaces $\mathbb{F}_n$
We have $\mathbb{F}_0= \mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1$ and, for $n \geqslant 1$, the $n-$th Hirzebruch Surfaces $\mathbb{F}_n$ is defined as a $\mathbb{C}\mathbb{P}^1$-bundle ...
- 2,430
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0
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77
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Finite dimensional space of vector fields
Consider a tangent bundle $TM$ and the space of sections $\mathfrak{X}(X)$. In general, for a $C^\infty$ structure, this space is an infinite dimensional vector space over $\mathbb{R}$. I'm trying to ...
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58
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Let $z=a e^t$ then $dz=ae^tdt$, How to define $d \overline{z}=?$
Let $z$ and $t$ be two complex variables such that $z=a e^t$ where $a$ is a (real or complex) constant. Thus, the differential form $dz$ is nothing but $ae^tdt$, i.e., $$dz=ae^tdt.$$ The question is:
...
- 2,430
2
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1
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64
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Extension theorem in $\mathbb{C}^n$
Suppose $\Omega\subset \mathbb{C}^n$ is a nonempty open set and $V\subset \Omega$ is a complex analytic (closed) manifold. Let $f$ be a holomorphic function on $\Omega \backslash V$ and for every $p\...
- 1,335
1
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1
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153
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The Limit Set of a Cyclic Group Generated by an Irrational Rotation and Acting on the Complex Projective Line
Let $\gamma$ be an irrational rotation (elliptic element) of ${\rm PSL} (2,\mathbb{C})$, $\Gamma$ the group generated by $\gamma$ $\left({\rm i.e.,} \Gamma= \langle \gamma \rangle \right)$ and $\...
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Question about complexified tangent bundle
In his book, Daniel Huybrechts define the complexified tangent bundle as: $T_{\mathbb C}U:= TU \otimes \mathbb C$.
But I don't understand this tensor product, I understand what $T_xU \otimes \mathbb C$...
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39
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Pulling-Back a Current by a Holomorphic Proper Non-Submersion Function Between Two Compact Complex Surfaces
Let $M,N$ be two Compact Complex Surfaces (compact complex manifolds of complex dimension $2$).
Let $A$ be a (non-empty) subset of $M$ (not necessarily a sub-manifold).
Let $f: M \longrightarrow N$ be ...
- 2,430
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81
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How to Push-Forward Differential Forms on a Complex Surface to a Complex Surface by a Holomorphic Function
Let $M$ and $ N$ be two Compact Complex Surfaces (i.e., $M$ and $N$ are Compact Complex Manifolds of complex dimension Two).
Let $A$ be a (non-empty) subset of the complex surface $M$ (not necessarily ...
- 2,430
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0
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Under what conditions can we push-forward the differential forms on a manifold $M$ to differential forms on a manifold $N$ by a smooth map $f$?
Let $M,N$ be two Complex Manifolds of the same (complex) dimension.
Let $f: M \longrightarrow N$ be a Smooth Map.
It is well-known that differential forms on the manifold $N$ can always be pulled-back ...
- 2,430
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40
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The Support of Differential Forms defined by Pulling-Back on Complex Manifolds of the Same Dimension
Let $M,N$ be two complex manifolds of the same (complex) dimension.
Let $\omega$ be a differential form on the manifold $N$ whose support denoted by $S$.
Let $f: M \longrightarrow N$ be a smooth map.
...
- 2,430
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0
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48
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Pullig-Back Differential Forms by Mappings Almost Diffeomorphisms
Let $M,N$ be two complex manifolds of the same (complex) dimension.
Let $A$ be a subset of $M$ (not necessarily a sub-manitold).
Suppose $f: M \longrightarrow N$ is a map such that $f: M \setminus A \...
- 2,430
0
votes
0
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27
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Is growth type of leaves of foliations invariant under diffeomorphisms?
Let $M$ and $N$ be two complex manifolds of complex dimension $2$.
Let $\mathcal{F}_M,\mathcal{F}_N$ be a singular holomorphic one-dimensional foliation on $M,N$; respectively. Thus, the leaves of the ...
- 2,430
5
votes
1
answer
90
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Why is Kummer's surface a smooth manifold
The $\mathbb{Z}/2\mathbb{Z}$ action on $\mathbb{T}^4$ has $16$ fixed points. Blowing up at these $16$ points, we get a new manifold $\tilde{\mathbb{T}}^4$. The action extends trivially at each ...
- 177
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0
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29
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Poincaré-Dulac Normal Form for Holomorphic Foliations of Complex Manifolds of Complex Dimension $2$
Let $\mathcal{F}$ be a singular one-dimensional holomorphic foliation on a complex manifold $M$ of complex dimension $2$ and $p \in M$ a singular point for the foliation $\mathcal{F}$.
Assume that the ...
- 2,430
1
vote
1
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108
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Slanted n-spheres and complex tori
$\newcommand{\C}{\mathbb{C}}\newcommand{\S}{\mathbb{S}}\newcommand{\Z}{\mathbb{Z}}$Complex tori, $\C^d/\Lambda$, where $\Lambda$ is a lattice in $\C^d$, generalise $\S^1\times\cdots\times\S^1$, by ...
- 263
1
vote
1
answer
89
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Characterization of harmonic $(1,1)$-forms
Let $(X,\Omega)$ be a compact Kähler manifold. Then there is the "usual" definition of the vector space $\mathcal H^{p,q}_{\bar\partial}$ which is the space of $\bar\partial$-harmonic $(p,q)$...
- 1,831
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1
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381
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Cohomology of pushforward and higher pushforward sheaves
I am reading through Peters-Steenbrink Mixed Hodge structures and I am having some trouble understanding what definitions of cohomology and hypercohomology are getting used. Let $U$ be a complex ...
0
votes
1
answer
400
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How is the real projective plane $\mathbb{RP}^2$ related to the complex projective line $\mathbb{CP}^1$?
I was looking for an intuitive understanding of complex projective spaces.
I picture $\mathbb{RP}^2$ as a square( $\mathbb{I}^2$, where $\mathbb{I}$ is a unit interval in $\mathbb{R}$ ) glued ...
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